The problem asks to determine whether the line segments $AB$ and $CD$ are congruent after plotting four points: $A(-2, 1), B(2, 1), C(3, 2), D(-1, -2)$ on a coordinate plane.

Steps:

1. Distance formula: To compare the segments, we need to calculate their lengths. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

2. Length of $AB$: $A(-2, 1), B(2, 1)$ Using the distance formula: $d_{AB} = \sqrt{(2 - (-2))^2 + (1 - 1)^2} = \sqrt{(2 + 2)^2 + 0^2} = \sqrt{4^2} = \sqrt{16} = 4$

3. Length of $CD$: $C(3, 2), D(-1, -2)$ Using the distance formula: $d_{CD} = \sqrt{(3 - (-1))^2 + (2 - (-2))^2} = \sqrt{(3 + 1)^2 + (2 + 2)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.66$

Conclusion:

• $AB = 4$
• $CD \approx 5.66$

Since the lengths of $AB$ and $CD$ are not equal, the segments are not congruent.

Would you like more details on any of the steps or further clarification?

Here are 5 related questions to explore:

1. What is the midpoint formula, and how can it be applied to line segments?
2. How would the graph look if we rotated it around the origin?
3. What other ways can we prove the congruence of two segments besides distance?
4. How can we determine if two line segments are parallel?
5. What is the significance of slopes in determining segment relationships?

Tip: The distance formula is essentially an application of the Pythagorean theorem in coordinate geometry.